of the Maritime University of Szczecin
Akademii Morskiej w Szczecinie
2017, 50 (122), 45–51ISSN 1733-8670 (Printed) Received: 24.02.2017
ISSN 2392-0378 (Online) Accepted: 08.06.2017
DOI: 10.17402/215 Published: 16.06.2017
Route prediction for a person in water drifting
in chosen basins using graph theory
Monika Kijewska
Maritime University of Szczecin, Institute of Marine Technologies
1–2 Wały Chrobrego St., 70-500 Szczecin, Poland, e-mail: m.kijewska@am.szczecin.pl
Key words: route prediction, trajectory prediction, object movement, search planning, survivor, person in
water, search and rescue, graph theory
Abstract
In this paper, the route prediction for a person in water was performed on the basis of a developed graph algo-rithm. This person drifted in water under the influence of surface currents and wind. The total drift route for the person in water was established as the route in a weighted directed graph. Vertices of this graph correspond to given points within a given basin. Additionally, the graph’s edges show possible directions of the overall human drift. The weight of the given edge describes the difference between the gradient of the edge and the total drift direction calculated on the basis of surface current field data and wind field data. An application has been created on the basis of a given algorithm which might be used to support the search for survivors in coastal areas (e.g. port basins, basins adjacent to the port, bays and sea areas) for which hydrodynamic models reliably reflect local phenomena.
Introduction
Search area prediction models for different objects in ocean waters are based on the Internation-al Maritime Organization (IMO) regulations which are presented in the International Aeronautical and Marine Search and Rescue Manual (IAMSAR) (ICAO & IMO, 2013). However, the methods pre-sented in this manual may indicate an incorrect or too large search area in coastal waters. This is due to the fact that the methods of finding the datum posi-tion in these areas takes sea currents into account, that is, the main, large-scale flow of ocean waters caused by, among other things, large-scale winds; therefore, the methods presented in this manual are based on the natural phenomena occurring in large water reservoirs with characteristics of ocean basins. Sea currents less than 25 nautical miles (Nm) dis-tance from the shore and less than 300 feet (100 m) water depth, and wind currents less than 20 Nm distance from the shore with water depth less than 30 m, are not taken into consideration. IAMSAR
recommends using data from short-term, reliable weather forecasts in coastal areas. These forecasts can be generated by hydrodynamic numerical mod-els with high resolution.
Solving the problem of determining different objects’ search areas requires considering, among other things, the following aspects: determining an accident area or an area where a drifting object has been observed and predicting an area where an object will drift to after a certain time period. The issue of determining an accident area or an area where a drifting object has been observed was solved by Pyrchla (2008). This issue was solved on the basis of fuzzy set theory. An accident area was established on the basis of a visual observation stream. This area was divided into subareas with a membership func-tion. This function describes the extent (between 0 and 1) to which the observer is convinced that the object belongs to a chosen subarea. The subareas were obtained by discretization of an accident area. As a result, a square grid was created. In each such subarea, one can assign a total current vector, which
affects an object located in this area, e.g. by using hydrodynamic models. Moreover, in each such sub-area we might define certain parameters of wind, which affect the object located in this area by using meteorological data or data generated by weather forecasting numerical models. In turn, for each sub-area, a center can be established. It is assumed that the subarea will move in the same manner as its cen-ter; therefore, this subarea may be identified by its center. An object’s total drift route needs to be estab-lished from the points which describe the routes of the subareas’ centers for an accident area or an area where a drifting object has been observed.
In this paper, a person in water route prediction algorithm is introduced. The person in water (PIW) moves under the influence of surface currents and wind in chosen basins. Furthermore, the perfor-mance of this algorithm is analyzed. The following basins are considered: port basins, basins adjacent to the port, bays, restricted areas and sea areas, where numerical models reliably reflect local phenomena. Possible PIW movement routes are modeled as the weighted directed graph routes. The subareas’ cen-ters, among which a PIW may move, are identified by graph vertices. The line segments joining the subareas’ centers, that are routes where a PIW may move, are defined by the graph edges. Weights are assigned to the edges of the graph. The edge weight describes the difference between the gradient of this edge and the direction of the PIW total drift, asso-ciated with the outgoing vertex of the edge. A PIW will move from one vertex to another vertex of the graph along the minimum-weight edges. The algo-rithm presented in this paper may be implemented and used to locate survivors when planning search and rescue operations within the chosen basins. Input data characteristics for an object’s total drift route prediction model
Location and characteristics of the study area
The Gulf of Gdańsk was chosen as the study area. It is located in the southeastern part of the Baltic Sea, between Poland and Russia. The Gulf of Gdańsk, which has a semicircular shape, sinks into the south-ern Baltic shore, connecting the Taran Cape (54°58’ N 19°59’ E) and Rozewie Cape (54°50’ N 18°20’ E). The entry width is 58 Nm, and the entry depth in land is 36 Nm. The Vistula Lagoon connects direct-ly with the waters of the bay by the Pilava Strait in the southeastern part of the bay, behind the Vistula Spit. The northwestern part of the Gulf of Gdańsk,
bordered by the Hel Spit from the northeastern side, is called the Bay of Puck (BHMW/HOPN, 2009).
Polish Marine Waters are characterized by, among others, a large variability in surface current directions (Pyrchla, 2008). Results from previous research indicate that the southern Baltic Sea region sea currents have different characteristics than those in large ocean basins (Pyrchla, 2008). A current’s direction and speed are determined mainly by wind direction and speed, and in the Vistula River Mouth regions they also depend on the currents in its arms (BHMW/HOPN, 2009). Wind distribution depends mainly on the atmospheric circulation. In the coastal area, the local conditions have an impact on wind direction and speed (BHMW/HOPN, 2009).
Surface currents as input data for a PIW route prediction algorithm
The directions and speeds of surface currents, achieved at the chosen points of the basin at given time points, are input data for the designed PIW route prediction application. The points of the basin create a regular grid. Hydrodynamic models generate these data. In Polish Marine Waters, two operational sys-tems are used: HIROMB (Funquist & Kleine, 2007) and M3D (Kowalewski, 1997). These models have the same reliability (Pyrchla & Kowalewski, 2009). Data used in this research was generated by the M3D model due to its higher resolution.
The operational hydrodynamic M3D model, created by the University of Gdańsk (UG) Ocean-ographic Institute, is characterized by high spatial resolution. In the special version of the M3D model – for the western part of the Gulf of Gdańsk – the spatial resolution of the discretization grid equals 0.1 Nm (185 m). In turn, the operational hydrody-namic HIROMB model, created by The Swedish Meteorological and Hydrological Institute (SMHI), is characterized by a spatial resolution of 1 Nm (1852 m).
The surface currents field forecast, presented in Figure 1, is used as input data for a graphical model. This forecast was generated by the M3D model for
July 2nd 2012 at 2000UTC. The PIW drift route
pre-diction starting points are indicated by the blue dots. The average surface current vector direction is south with an average speed of approximately 3.6 cm/s.
The data on surface currents fields as the input data for the graphical model will be updated. Data on surface currents fields at 2100UTC, 2200UTC were also generated by the M3D model. It turns out that over time the average surface current vectors’
direction change, becoming SSW and SW respec-tively. The average surface current speeds also increased: in the area around the blue dots present-ed in Figure 1, the forecastpresent-ed average surface cur-rent speed was equal to approximately 9.7 cm/s at 2100UTC, and approximately 10 cm/s at 2200UTC.
Wind characteristics received from IMGW weather stations
Data on wind directions and speeds are also input data for the created PIW route prediction application. These data were received for the Institute of Meteo-rology and Water Management (IMGW) meteo sta-tions. The following meteo stations were taken into consideration: Gdynia, Gdańsk – Northern Port and Hel. In the Gdynia meteo station, measurements are collected in six-hour time intervals. The wind direc-tion is expressed as one of the 16 compass points.
For July 2nd 2012 at 1800UTC, the following
aver-age wind parameters were achieved: NNE direction, 3 m/s speed. For the Gdańsk – Northern Port and Hel stations, measurements were collected in
one-hour time intervals. For July 2nd 2012 at 2000UTC,
2100UTC, 2200UTC the average wind directions and average speeds are given in Table 1.
To establish the average wind direction and speed
for July 2nd 2012 at 2000UTC, 2100UTC, 2200UTC
for the western part of the Gulf of Gdańsk, the achieved measurements for the Gdańsk – Northern Port and Hel meteo stations were taken into consid-eration. Further, the calculated average wind para- meters were compared to the Gdynia meteo station measurement at 1800UTC. The average wind direc-tions and speeds are given in Table 2.
The established average wind directions and speeds correspond to the wind measurements for the
Gdynia meteo station at 1800UTC: 3 m/s in the NNE direction. The averages presented in Table 2 were input to the application.
PIW leeway characteristics for the western part of the Gulf of Gdańsk
Wind affects the part of the PIW that is immersed causing leeway of the PIW. The leeway depends on the PIW’s position in the water: Sitting PIW, Hori-zontal PIW or Vertical PIW. For the Gulf of Gdańsk area, which is characterized by low wind speeds, according to Allen and Plourde (Allen & Plourde, 1999) a PIW’s leeway direction is the same as the wind direction; therefore, its speed may be estab-lished as follows: 54.557 54.556 54.555 54.554 54.553 54.552 54.551 54.550 54.549 Latitude [deg N] Longitude [deg E] 18.56 18.565 18.57 18.575 18.58 18.585 18.59
Figure 1. Sample forecast of the surface currents field (July 2nd 2012, 2000UTC)
Table 3. Leeway directions and speeds for July 2nd 2012 for
the western part of the Gulf of Gdańsk
PIW Position [UTC]Hour direction Leeway [°] Leeway speed [m/s] Vertical PIW 2000 –127 0.2351 Vertical PIW 2100 –153 0.22925 Vertical PIW 2200 –119.5 0.2351 Sitting PIW 2000 –127 3.815 Sitting PIW 2100 –153 3.8125 Sitting PIW 2200 –119.5 3.815
Horizontal PIW – Survival Suit 2000 –127 5.2932 Horizontal PIW – Survival Suit 2100 –153 5.286 Horizontal PIW – Survival Suit 2200 –119.5 5.2932
Table 1. Average wind directions and speeds for July 2nd
2012 for the chosen meteo stations
Meteo
station [UTC]Hour
Average wind direction [°] Average wind speed [m/s] Gdańsk – Northern Port 2000 53 4 2100 13 3 2200 50 4 Hel 2000 53 2 2100 41 2 2200 71 2
Table 2. Average wind directions and speeds for July 2nd
2012 for the western part of the Gulf of Gdańsk
Hour
[UTC] Average wind direction [°] Average wind speed [m/s]
2000 53 3
2100 27 2.5
Vertical PIW = 1.17%·W10m + 0.2 [cm/s] (1)
Sitting PIW = 0.5%·W10m + 3.8 [cm/s] (2)
Horizontal PIW – Survival Suit =
= 1.44%·W10m + 5.25 [cm/s] (3)
where W10m is the 10-meter wind speed (m/s).
The leeway vector forecast has been established based on the data in Table 2. The leeway vector parameters are presented in Table 3.
PIW route prediction algorithm description based on a graphical model
A PIW’s total drift route is presented as a route on a weighted directed graph. The PIW will drift along a route which is a sequence of edges. More-over, this route is achieved by minimizing the differ-ence between the PIW’s total drift direction derived from the graphical model, and its total drift direction derived from the data generated by hydrodynam-ic models and the measured data provided by the IMGW, over certain time intervals.
The ordered triple (V, E, f), where V ≠ Ø, E ⊆ V 2
and f : E → R, is called the weighted directed graph
G. The set V is the vertex set of the graph G. The
elements of the set V are the vertices of the graph
G. The set E is the edge set of the graph G. The
ele-ments of the set E are the edges of the graph G. Let
ij be an edge of the graph G. ij is called the outgo-ing edge of the vertex i and the outgo-ingooutgo-ing edge of the
vertex j. The function f is the weight function of the graph G. The real number w = f(ij) for ij ∈ E is the
weight of the edge ij.
In work by Kijewska (Kijewska, 2016), the con-cept of the weighted directed graph definition is pre-sented. This graph describes the surface currents’ interaction on the object immersed in water so that the part extending above the water is not larger than the human head. The considered basin is discretized. As a result of the discretization, a regular grid of nodes is achieved. These nodes are the points numbered consecutively from 1 to i in this basin. The location data are associated with the considered grid nodes by using geographical coordinates. In addition, surface currents’ directions and speeds, generated by a hydro-dynamic model, are also associated with those nodes. The vertex set V of the graph G is defined as the set
V = {1, 2, 3, …, i} (4)
where the vertex i corresponds to the i-th node of the discretization grid. Moreover, the data assigned to the grid nodes may be associated with the vertices of
the graph. The ordered pair of the vertices i, j is the edge ij in the graph. This edge exists in the graph G, if the distance between the nodes i and j of the grid is not greater than some fixed number r > 0. Select-ing the number r is described by Kijewska (Kijew-ska, 2016). The edges of the graph are routes, along which the PIW may move under the influence of surface currents affecting it. A weight is associated with each edge of the graph. This weight describes the difference between the gradient of this edge and the direction of the surface current assigned to the vertex, from which this edge extends.
It is worth noting that the wind affects the part of the PIW immersed under water. The strength of this effect depends on the PIW’s position in the water – sitting, horizontal or vertical. The wind causes a lee-way of the PIW. The PIW’s total drift is computed by adding the surface current and the leeway.
Let the weighted directed graph G = (V, E, f) be determined as in Kijewska (Kijewska, 2016), but the weight function f : E → R be determined as follows:
180 , 360 180 , b b b b ij f (5)where: b = |α – βi|, α – the gradient of the edge ij
of the graph G, βi – the direction of the PIW’s total
drift established at the vertex i. Therefore, the edge weight of the graph G may be described as the differ-ence between the gradient of this edge and the direc-tion of the total drift associated with the vertex, from which this edge extends.
Recalling that the centers of all the subareas within an accident area or an area where the PIW has been observed will be moved. Next, a node of the discretization grid located closest to the place where the PIW is should be identified. If the distance between this node and the PIW’s position is too great, then the bilinear or nearest neighbor interpo-lation of the grid should be used. The node, located closest to the place where the PIW is, corresponds to a certain vertex of the graph G. The PIW will move from this vertex along the outgoing edge with the minimum-weight. The sequence of edges, where the PIW will move, creates its drift route.
The PIW drift prediction algorithm has been writ-ten in SCILAB and consists of the following steps:
1. the discretization of the considered basin; in addition, for each node of the discretization grid, information about its longitude and latitude is included;
2. loading the data (surface currents’ directions and speeds and wind directions and speeds for the chosen basin over a fixed time point);
3. establishing surface currents’ directions and speeds and wind directions and speeds at all of the nodes of the grid (e.g. by their bilinear or nearest neighbor interpolation);
4. establishing the geographical coordinates
(φp; λp) for the starting point p of the prediction
(the starting point p is the grid point);
5. entering the time ta for updating surface currents
and wind data (this value may be established by the application based on the information about time intervals downloaded with new data);
6. entering the prediction time tpr;
7. determining the vertex set V of the graph G (recall that the vertex i of the graph G corre-sponds to the i-th node of the grid);
8. calculating the leeway direction and speed at node p of the grid;
9. calculating the PIW’s total drift vector direction and speed at node p of the grid;
10. calculating the radius r of the circle encompass-ing the vertices of the graph G beencompass-ing the neigh-bors of the vertex p;
11. defining the edge set E of the graph G; 12. calculating the edge weights of the graph G; 13. selecting the minimum-weight extending from
vertex p;
14. examining whether this weight is not greater than the allowed PIW route prediction error for one step. If this weight is greater than this error, then one should use bilinear or the nearest neigh-bor interpolation of the discretization grid and the data associated with it (such interpolation allows for the consideration of more PIW move-ment directions from a given vertex of the graph and decreases movement time between vertices of the graph) and return back to step 7;
15. moving the PIW along the minimum-weight edge to the new vertex j of the graph G;
16. calculating the required time t of the PIW move-ment along this edge by the formula t = s/V, where s – the PIW route, V – the total drift speed associated with the vertex p; in addition, the times required to move along the chosen edges of the graph are summed;
17. examining whether the time simulation tpr is
achieved or the vertex , where the PIW is locat-ed, is situated on land; if at least one of these conditions is satisfied, then the model termi-nates and the results are displayed on the screen (the predicted PIW drift route) and these route parameters should be saved in the file;
18. examining whether the data updating time ta is
achieved. If this condition is satisfied, then the
data on the surface currents and wind are updat-ed, the edge weights of the graph are re-calculat-ed. Next, the algorithm performs steps 7 and 8, but the vertex j (see step 15) is used in place of the vertex p when step 13 is performed.
Experimental verification of the graphical model algorithm
To examine the PIW’s total drift route prediction accuracy based on the described graphical model, data from two real experiments are used. One of them was based on throwing a buoy into the water and tracking its total drift route by GPS. The buoy was constructed in such a way that it imitated a Ver-tical PIW. A second experiment was based on throw-ing a PIW clad in a survival suit (PIW/SS) into the water and tracking its total drift route. The PIW/SS floated in a horizontal position on the water’s
sur-face. These experiments were carried out on July 2nd
2012 at 2008UTC. The buoy was thrown into the Gulf of Gdańsk waters at longitude 18.573969° E and latitude 54.555896° N. In turn, the PIW/SS was thrown in at longitude 18.572973° E and latitude 54.556061° N.
Furthermore, computer simulations were carried out. From these simulations, the forecasted total drift routes of both the buoy and PIW/SS were calculated. These routes were computed on the basis of the pre-sented graphical model.
Real and simulated routes of the PIW clad in a survival-suit
In Figure 2, the following routes are presented: the real total drift route of the PIW/SS measured by GPS (black color) and the simulated route based on the graphical model (blue color). Moreover, for
54.557 54.556 54.555 54.554 54.553 54.552 54.551 54.550 54.549 Latitude [deg N] Longitude [deg E] 18.56 18.565 18.57 18.575 18.58 18.585 18.59
Figure 2. Route prediction of the PIW/SS (black – route of the real-life PIW/SS, blue – simulated route of the PIW/SS)
each of the routes, two points have been indicated. For each of the routes, the first point indicates the location where the PIW/SS was at 2100UTC. The second one is the point, where the PIW/SS was at 2200UTC. Each of these points has been marked as a white point on the routes. During the real experi-ment, it was been measured by GPS that at 2100UTC the PIW/SS was located at longitude 18.570126° E, latitude 54.553486° N and at 2200UTC at longitude 18.566206° E, latitude 54.551537° N. When the simulated experiment was carried out on the basis of the devised graphical model, the PIW/SS was locat-ed at longitude 18.571032° E, latitude 54.553215° N at 2100UTC and at longitude 18.568175° E, latitude 54.550358° N at 2200UTC. The distance between the real location of the PIW/SS and the forecasted location equaled approximately 70 m at 2100UTC and approximately 185 m at 2200UTC.
Real and simulated routes of the buoy
In Figure 3, the following routes are present-ed: the real total drift route of the buoy measured by GPS (black color) and the simulated route based on the graphical model (blue color). Moreover, for each of the routes, two points have been indicat-ed. For each of the routes, the first point indicates the location where the buoy was at 2100UTC and the second one is the point where the buoy was at 2200UTC. Each of these points has been marked in white on the plotted routes. During the real exper-iment, it was measured by GPS that at 2100UTC the buoy was located at longitude 18.572227° E, latitude 54.553867° N and at 2200UTC at longitude 18.569405° E, latitude 54.551866° N. When the simulated experiment was carried out on the basis of the devised graphical model the buoy was located
at longitude 18.572957° E, latitude 54.5532° N at 2100UTC and at longitude 18.570699° E, latitude 54.550189° N at 2200UTC. The distance between the real location of the buoy and the forecasted loca-tion equaled approximately 90 m at 2100UTC and approximately 205 m at 2200UTC.
Conclusions
In this paper, a total drift route prediction algo-rithm for a person in water has been presented. This algorithm allows the prediction of the drift route in coastal basins. Furthermore, an experimental verifi-cation of the simulated person in water drift routes was carried out. It was observed that the person in water’s real and simulated drift routes achieved a good agreement (the direction of the forecasted person in water drift route is generally in accor-dance with the observed direction). One may then conclude that the devised application based on the graphical model is able to reliably determine the per-son in water’s total drift route. This has the effect of narrowing down the search area and decreasing the search time. It is also worth observing that the pre-sented model takes into consideration data on surface current fields, wind fields and leeway fields. Further-more, those data are updated during the simulation. This approach is dedicated to applying a greater use of data from hydrodynamic models and weather forecasting models during SAR operations. Current-ly, these models are being dynamically developed. The simulated results may be improved if the data on spatially-varying wind fields are entered from a reli-able weather forecast model.
Acknowledgments
This research outcome has been achieved under the research project No S/4/S/INM/15 financed by a subsidy from the Ministry of Science and Higher Education for statutory activities. Furthermore, the author would like to thank the Institute of Meteo-rology and Water Management (IMGW) for sharing the data on wind parameters for the chosen meteo
stations for July 2nd 2012. The author would like to
also thank the Oceanographic Institute UG (name-ly Marek Kowalewski, PhD) for sharing data on the surface currents’ parameters and the real experimen-tal data for the western part of the Gulf of Gdańsk
for July 2nd 2012. Moreover, the author would like to
thank Krzysztof Pleskacz, PhD, Eng., for providing useful comments. The author expresses their sincere gratitude to all of the above.
54.557 54.556 54.555 54.554 54.553 54.552 54.551 54.550 54.549 Latitude [deg N] Longitude [deg E] 18.568 18.569 18.57 18.571 18.572 18.573 18.574 18.575
Figure 3. Route prediction of the buoy (black – route of the real-life buoy, blue – simulated route of the buoy)
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